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Why there is a "tooth brush" in Transmon qubit?


One might naturally have this question in mind, once he saw the sketch of the transmon qubit above, that why is there a "Tooth Brush" like structure in the middle?



After I watch a amazing talk on youtube given by Alexandre Blais - Quantum Computing with Superconducting Qubits (Part 1) - CSSQI 2012,(https://www.youtube.com/watch?v=t5nxusm_Umk)

I finally understand what this structure is about!


In a nutshell, it is just a capacitor! The tooth brush structure is so designed that It can add up to the capacitance in the transmon qubit efficiently without taking more space on the quantum chip.


Then, another question comes out: Why do we need such a large capacitance for construting the transmon qubit? I will show you the notes that I've taken about the video. If you are interested, I recommend you to watch the video directly.


First, we should recall the basic structure of a quantum harmonic oscillator.




After we quantize the energy of a simple harmonic oscillator, we get a equally gapped energy band structure. This can not serve the purpose to build a qubit because we have to add anharmonicity. (When we drive a harmonic oscillator from |0>, the final state will be a coherent state.)


In transmon qubit, we add a non-linear part, which is the famous Josephson junction, and form the new LC-circuit. Now we calculate the energy by integration and get the constant Ej with a cosine term.

We rewrite the energy of capacitor and get another important number Ec. Ec is basically how much energy it takes to charge a cooper pair to the conductor.



Finally, we get the hamiltonion of the transmon structure. Now, the problem is how should we choose the best Ec/Ej ?

After observing the solution to the hamiltonion, we find that Ej/Ec should be large enough for the energy gap to appear. What's more, even there is energy gap, the system can be susceptible to dephasing because a minor change of charge due to environment noise can change the energy gap!


When Ej/Ec get larger, the energy curve become flat. That explain why we what a large capicitance! 1/Ec is actually propotional to the capacity value C in the transmon qubit.


However, Ej/Ec cannot be too large, when the whole system degrade to a simple hamonic oscillator. So we have to carefully calculate the optimized value of Ej/Ec.


This is the frst time I really undertand the tradeoff between "Anharmonicity" and "Robustness to noise".

A further question to think about is that : Is there any qubit structure which anharmonicity and noise robustness(Or energy dispersion) increase and decrease at the same time?

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